Gauss Hermite

Motivation

Gauss-Hermite quadrature is designed for integrals of the form \(\int_{-\infty}^{+\infty} f(x)\,e^{-x^2}\,dx\), which appear naturally in quantum mechanics, probability theory (Gaussian distributions), and signal processing. By exploiting the orthogonality of Hermite polynomials, it provides exponential convergence for smooth integrands with Gaussian-type decay.

Example

use integrate::gauss_quadrature::gauss_hermite_rule;

let f = |x: f64| 1.0;

let n:usize = 100;

let integral = gauss_hermite_rule(f, n);
println!("{}",integral);

Understanding Gauss-Hermite rule

Gauss-Hermite quadrature formulas are used to integrate functions \(f(x) e^{-x^2}\) from \(-\infty\) to \(+\infty\).

With respect to the inner product

\[ \langle f,g \rangle = \int_{-\infty}^{+\infty} f(x) \cdot g(x) \cdot w(x) , dx \]

the Hermite polynomials

\[ H_n(x) = (-1)^n \cdot e^{x^2} \cdot \frac{\partial^{n} e^{-x^2}}{\partial x^n} \quad \text{for} \quad n > 0 \]

and \(H_0(x) = 1\) form an orthogonal family of polynomials with weight function \(w(x) = e^{-x^2}\) on the entire \(x\)-axis.

The \(n\)-point Gauss-Hermite quadrature formula, \(GH_n ( f(x) )\), for approximating the integral of \(f(x) e^{-x^2}\) over the entire \(x\)-axis, is given by

\[ GH_n ( f(x) ) = A_1 f(x_1) + \cdots + A_n f(x_n) \]

where \(x_i\), for \(i = 1,\ldots,n\), are the zeros of \(H_n\) and

\[ A_i = \frac{2^{n+1} \cdot n! \cdot \sqrt{\pi}}{H_{n-1} (x_i)^2} \quad \text{for} \quad i = 1,\ldots,n \]

Limitations

Gauss-Hermite quadrature is only appropriate for integrals of the form \(\int_{-\infty}^{+\infty} f(x)\,e^{-x^2}\,dx\). If the integrand does not have Gaussian decay, the method will produce inaccurate results. It is not suitable for integrands on finite intervals or those with non-Gaussian tail behavior.